1. Use of interpolation and FSAL property for integration in field
Dmitry Sorokin, MIPT John Apostolakis, CERN

2. Content Introduction Part 1: New design of the propagation classes
Part 2: The Bulirsch-Stoer method
Part 3: Using FSAL in integration
Results
Future work: Using interpolation in integration
Sample tracks from Geant4 b2a example

3. Introduction Large number of track segments (steps) in each simulation
Robust integration of a variety of equations (with a user provided field)
High cost of field evaluation
Need to find the intersection point between a curved trajectory and the surfaces of the experiment

4. Part 1: New design of the propagation classes

5. Part 2: The Bulirsch-Stoer method
Idea:
Use midpoint method to estimate integral
Vary number of intermediate points
Approximate the integral using rational functions & extrapolate to n = ∞
Advantages:
Step size and order control
Very good for smooth problems and large steps
Can provide dense output
n = 2i+2, for i = 0,1,2….
n = 4i+2, for i = 0,1,2….

6. Part 3: Using FSAL in integration
Derivative at endpoint
(for successful step)
The decrease in the number of field evaluations when its FSAL property is used (the estimate of the final derivative is reused if valid), in comparison with its use when it’s FSAL property is not used.

7. Test: propagation in uniform magnetic field
1 MeV proton in the uniform 1 tesla magnetic field. Radius of the circle in xz plane is mm Momentum is MeV/c. delta1step = 1e-4 mm, deltaIntersection = 1e-5

8. Test: NTST
Main integral test to check the overall performance of the simulation which is used is test NTST. It simulates the inner detector of the Babar experiment, with the realistic interpolated Babar field. Sample events are sample Babar events using the Pythia 6 event generator.
File
range cut
looper cut
min
Ecat
run2a.mac
1 mm
200 MeV
1 MeV
run2b.mac
Not applied
run2c.mac

9. Next step: Using interpolation in integration
Old strategy
Make series of steps without error control to predict the step size (satisfy sagitta distance d < δchord)
Make a step with error control to improve the accuracy (ΔB < δ1step)
If the chord intersects:
Make a series of substeps with error control to locate the intersection point
New strategy
Make one step with error control
Use dense output to divide the step to substeps (satisfying d < deltachord)
For each substep:
If the chord intersects:
Use dense output to locate the
intersection point
Pros: a lot fewer field evaluations required for large steps
Cons: dense output is less accurate than the solution

10. Other new integration methods

11. New methods for step size control (Runge Kutta)
How to calculate the size of the next step after a successful step ?
New method – more stable (vs oscillations)
Old method
Standard driver
Gustafsson driver
Decrease, %
run2a.mac
178’346
178’340
.003
run2b.mac
635’086
627’125
1.25
run2c.mac
2’068’425
2’016’590
2.50
Number of calls to field per event in the NTST test. Stepper is g4dormandprince745

12. Summary Introduced FSAL method to reuse endpoint derivative
Alternative Estimation of Next Step size for RK steppers – small benefit
New methods (Bulirsch-Stoer) for Integration Provided
Prepared for use of Interpolation – a revolution in integration

13. Thank you

14. RK method adapted for oscillatory problems
Proof of principle: use method
adapted for oscillatory solutions based on classical RK4 method
G4ClassicalRK4
ARK4
Decrease, %
run2a.mac
259’111
237’066
8.5%
run2b.mac
1’251’644
1’047’696
16.3%
run2c.mac
4’283’088
3’565’924
16.7%
Test NTST, number of field evaluations per event for G4ClassicalRK4 stepper and ARK4 stepper (adapted for oscillatory solutions classical RK4 stepper)